SOLUTION: Among all pairs of numbers whose sum is 8​, find a pair whose product is as large as possible. What is the maximum​ product?

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Question 1102022: Among all pairs of numbers whose sum is 8​, find a pair whose product is as large as possible. What is the maximum​ product?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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Let x and y be the pair such that x + y = 8.


Then y = 8-x, and they want you find x and y in a way that the product x*y would be maximal.

x*y = x*(8-x) = -x^2 + 8x = -(x^2 -8x) = -(x^2 -8x + 16) + 16 = -(x-4)^2 + 16.


The right side is the parabola opened down in the vertex form


It has the maximum at x = 4,  and the maximum value is 16.


So, your answer is: the pair (x,y) under the question is the pair x= 4, y= 4, or (4,4),  and the maximum product is  4*4 = 16.


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On finding maximum of a quadratic form and associated problems see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
    - Using quadratic functions to solve problems on maximizing revenue/profit
    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.